Even though the notion of an exponential dichotomy is an important concept, there are many important classes of systems that do not fit in this framework. For example, Coppel [1]} gave an example
of a planar nonautonomous system without bounded growth that does not admit an exponential dichotomy even though its restriction to the stable and unstable subspaces are contractions and expansions, respectively. As observed in [2]}, what fails to be satisfied for this example to admit an (uniform) exponential dichotomy is that the amount of time that we need to wait until we see actual contraction (or expansion) depends on the initial time. In terms of condition (REF ) this means that the coefficient \(K\)  may depend on \(n\)  (or \(m\) ) and that it can go to infinite whenever \(n\)  (or \(m\) ) vary.
This gives rise to the notion of nonuniform exponential dichotomy introduced by Barreira and Valls [3]}, which turns out to be ubiquitous in the context of the ergodic theory.
